DEPARTMENT OF AVIONICS
DIGITAL TECHNIQUES (3 credit)
Course Code: AMEAV 303
1.0 Number Systems and Codes
Digital electronics is the electronic circuits that performs the operational task of a
logical decision making process. They are significantly different from other electronics
such as those used in old radios, television sets and oscilloscopes, all of which are
known as Ana-logue electronics.
DEFINITIONS OF DIGITAL AND ANALOGUE
Digital - Something of incremental nature changing in steps from one definite value to
another. In digital electronic devices, voltages and currents changes in step from one
definite value to another. Digital electronics provide for greater reliability, faster
response, smaller components, lighter equipment, and lower operating cost than can
be provided by analog systems. Digital systems increase the mean time between
failures and reduce the subsequent repair time for failed equipment.
Digital aircraft systems contain built-in test equipment (BITE) that provides rapid
fault isolation. The majority of digital aircraft systems contain several line replaceable
units (LRUS). Defective LRUS may be quickly identified by BITE system and
exchanged during ground maintenance. Use of the LRU and BITE concepts greatly
reduces aircraft maintenance down time.
Another concept of digital aircraft technologies is to remove as many moving parts
from the electrical system as possible. Through out the aircraft switches are replaced
with proximity indicators, relays are replaced with transistors, and instruments are
replaced with digital displays.
Analogue - Something continuous which changes in a continuous manner. In analogue
electronic devices, voltages and currents changes in a continuous manner.
Two examples of devices that can be either analogue or digital are clocks and
voltmeters. The minute and second hands of analogue electric clock move
continuously, whereas on a digital clock the numerals changes abruptly at the end of
every second and minute. Likewise, an analogue voltmeter has a meter face with a
needle that can move gradually from end of the scale to the other whereas on a digital
voltmeter the voltage is displayed in discrete digits each of which changes abruptly.
Digital and Ano-logue wave forms - Fig 1.1 and 1.2 below shows a circuit and
corresponding waveform of a digitally changing shape, while fig. 2.1 1 and 2.2 shows
an analogue circuit and a sample of gradually changing shapes (Analogue).
DIGITAL NUMEROLOGY
In computer Arithmetic, there are four number systems:
1.
Decimal number system in which we count from 0-9.
2.
Binary number system in which we count from 0-1 or in powers of 2
3.
Octal number system in which we count from 0-7 or in power of eight.
4.
Hexadecimal number system in which we count from 0-15 or in powers of
sixteen.
DECIMAL NUMBER SYSTEM
We are used to the decimal number system in which we count from 0 to 9. This may
well have originated from finger counting on ten fingers. The decimal number system
has ten different basic stages or digits, 0 through 9, thus having a base of 10.
Counting above 9
When we want to count above 9 in the decimal number system, we combine two or
more of the basic digit to express any number we need. Thus, with the basic digits we
can encode numbers up to 10, 1000, and larger.
Digit Position
Each digit position in the decimal number system represents a power of ten: when we
write a number such as 5738, we can represent it as follows:
103
102
101
100
1000
100
10
1
5
7
3
8
+
(3 x 10)
In other words, we mean:
(5 x 1000)
+
(7 x 100)
+
(8 x 1) =
5738
Each digit position in the above diagram represents a power of ten. The number ten is
known as the base or radix of the decimal system. Therefore, decimal number can be
defined as the one in which we count in powers of ten.
BINARY NUMBER SYSTEM
The binary number system uses two digits 0 and I and has a base of two.
Counting above 1
When we want to count above 1 in the binary number system, we combine two or more
of the basic digits to express any number we need. Thus, with the two basic digits we
can encode a large number. One common way of expressing numbers in the binary
number system is as follows:
Digit Position
Each digit position of the binary number system represents a power of 2. When we
write a number such as 1001 (binary) we can represent it as follows:
23
22
21
20
8
4
2
1
1
0
0
1
(0 x 2) +
(1 x 1) =
1 In other words, we mean:
(1 x 8) +
(0 x 4) +
9
That is 10012 = 910 or 1001 (binary) = 9(decimal)
Each digit position in the above equation represents a power of two. The number two
is known as the base or radix of the binary number system. Therefore, binary-number
system can also be defined as the one which we count in powers of two.
The above is referred to as the binary code and can be used to encode numbers
larger than 31 by simple adding digits that represent powers of 2 higher than 2a.
The number 351 is encoded as 101011111.
Assignment - convert the following binary numbers to decimal:
l.
1011
2.
1100
3.
10010
4.
11001
5.
101101
DECIMAL - TO - BINARY CONVERSION
One method for converting from decimal to binary is to use a process of successive
division of the decimal number by the binary base (2). The remainder of each of
these divisions is marked down and becomes a part of the binary number, example
- Convert the decimal number 688 to its equivalent binary coded number.
Read Binary number upward = 1010110000
688 = 10101100002 or
688 Decimal = 10 110000 Binary.
Assignment - Convert the following decimal numbers to binary:
1. 351
2. 86
3. 264
4. 144
CONVERSION OF FRACTION;
All binary numbers appearing to the right of binary decimal point represents
a fraction. The order is descending from the right.
Assignment: Convert the following binary fractions to decimal fraction:
1. .011
2. .1001
3. .110
4. .0011
5. .1101
Conversion of decimal fraction to binary fraction
Decimal fraction may be converted to binary fractions by successively
multiplying the decimal fraction by 2. After each multiplication the inter
portion is removed to form the binary fraction by building to the right of the
binary decimal point,
Example 1 - Convert 0. 125 decimal to binary
0.125
2
0.250
2
0.500
2
1.000
2
.000
The process may be continued to achieve the required accuracy.
Example 2 -
Convert .029,
o to a five digit binary fraction.
0.29
2
0.58
2
1.16
2
0.32
2
0.64
2
28
Assignment - Convert the following decimal fractions to binary fraction:
1
.625
2.
.825
3.
.325
4.
.125
5
.425
BINARY ADDITION
The rules for adding binary numbers are basically the same as for any other number
system. since each column of digits represent a power of 2 a carry to the next most
significant column occurs if the result of adding a column is grater than or equal to 2.
We have already seen that the only numbers used in the binary system are 0 and l.
This enables us to formulate very simple and specific rules for binary addition.
Rules for addition of binary numbers:
0
+
0
=
0
0
+
1
=
1
1
+
0
=
1
1
+
1
=
10
Apply this rule and add 5 + 7 = 12
First convert to binary 0101 = 5
0111 =
7
1100 = 12
Assignment. Carry the following additions:
l.
8
+
9
2.
13
+
16
3.
7
+
13
4.
15
+
10
5.
21
+
14
BINARY SUBTRACTION Rules for subtraction of binary numbers (Direct
subtraction):
0 -0 = 0
1 -0 = 1
1- 1 = 0
10- 1 = 1.
Example: subtract 33 - 11 = 22
Convert to binary:
33
=
100001
11
=
001011
10110
BINARY MULTIPLICATION - The following rules are used for multiplication of
binary numbers:
0
x
0
=
0
0
x
1
=
0
1
x
0
=
0
1
x
1
=
1
BINARY DIVISION - The following rules are for division of binary numbers:
0
/
0
=
0
0
/
1
=
0
1
/
0
=
(infinity)
1
/
1
=
1
Assignment: Carryout the following subtractions:
1.
12 - 8
2.
9 - 4
3.
24 - 14
4.
15 - 8
5.
36 - 24
COMPLEMENTS - The complement of a number is the difference between the power
of the base of the complementation and the number being complemented. These are
used in digital computers for simplifying subtraction operation and for logical
manipulations.
Base of complementation
Number being complemented
The 10's complement of 2 is
8 (base 10-2)
The 9's complement of 3 is
6 (base 9 - 3)
The 2's complement of 1 is
1 (base 2 - 1)
The 1's complement of 1 is
0 (base 1 - 1)
1's COMPLEMENT - Because the binary system has only two states the 1's
complement of a binary number can be obtained simply by writing every bit in its
opposite state; i.e. every 1 bit is changed to 0, and every 0 bit is changed to 1.
Examples:
1's complement of 11011 is 00 100
1's complement of 01010 is 10101
2's COMPLEMENT - The easiest method of finding 2's complement of any binary
number is to first obtain the 1's complement, then add 1 as follows:
001101
Number
110010
1's complement of the number add 1
+1
110011
2's complement of the number
Assignment - Find 1's and 2's complement of the following:
1.
00111
2.
11000
3.
010101
4.
101010
5.
11001100
BINARY CODED SYSTEMS
General Theory
There are several varieties of code systems used to convert information bits into letters
or decimal numbers. The pure binary number system is often clumsy for a computer
to manipulate data quickly. Three of the more common systems used to provide faster
computers are called the BINARY CODE DECIMAL, OCTAL NOTATION and
HEXADECIMAL systems. All of these systems utilize the binary digits 'I' and `0' to
represent decimal or base 10 numbers.
Binary-Coded Decimal system
The binary coded decimal (BCD) system uses a group of four bits to represent each
digit of a decimal number. Each digit of a decimal system (0 to 9) can be represented
by four binary digits. For example `9' in decimal system equal 1001 in binary system.
This system is extremely useful when dealing with large quantities of data exchanged
between the inputs and outputs of a computer system.
Another advantage of a BCD is that there are extra units unused by the four digit
system. Since `9' (1001) is the highest decimal number to be represented by the four
bit binary code, there are six unused combinations for each byte. These additional
digits are often assigned various alphabetic symbols, thus expanding the information
carried by each four bit byte. below is a table showing BCD and their decimal
equivalents.
OCTAL NOTATION SYSTEM
Octal notation system:is a binary representation of an octal number. Octal numbers
are composed of eight different symbols; octal notation uses 'I' and `0'. Octal notation
is composed of a series of three-bit groups (triad group). Since the largest decimal
number represented by three binary digits is 7(111), this is a base 8, octal system. In
short, octal notation is a means to represent octal (base 8) number in binary language.
Octal notation is useful for certain programming techniques where large quantities of
binary numbers must be manipulated. Octal notation is often used for the transmission
of data by aircraft computers and their related peripherals.
DIGIT POSITION - Each digit position in the octal number system represent a power
of
8. When we write a number such as 235 octal we can represent it as follows:-
Each digit position in the above diagram represents a power of eight. The number eight
is known as the base of the octal number system and can also be defined as one in
which we count in powers of eight.
OCTAL-TO-DECIMAL CONVERSION - The following procedure can be used to
convert octal numbers to decimal equivalents: Convert 230 (decimal) to octal.
2308 =
2(82) +
3(81) +
0
(80)
=
2(64) +
3(8)
+
0
(1)
=
128
24
+
+
0
= 15210
Assignment:- Convert the following octal numbers to decimal:1)
462
2)
84
3)
534
4)
138
5)
38
DECIMAL - TO OCTAL CONVERSION
As in the decimal - to-binary conversion, decimal - to octal conversion requires the
base 8 to be repeatedly divided into the decimal number to obtain an octal equivalent.
The octal number is obtained by reading the remainder column upwards
BINARY-TO-OCTAL-CONVERSION - The following steps are followed to
converting binary to octal: e.g convert the binary number 10100111011 to octal.
1.
Divide the numbers into groups of 3 bits (triads) starting at the least significant
bit, i.e. the right hand side. The above number is then written as:
10 100 111 011
2.
Now convert each group of 3 bits to the equivalent decimal numbers.
i.e.
1 100 111 011 becomes 2 4 7 3
Put the converted digits together to give the equivalent octal number i.e. 2473.
Assignment - convert the following binary numbers to octal number:
1)
1011101110001
2)
11000100011101
3)
10101110010110
OCTAL-TO-BINARY CONVERSION - to convert from octal-to- binary we reverse
the rules above e.g. To convert the octal number 3072 to binary we write down the
binary code for each digit in the octal number i.e.
3072
becomes 011 000
111
010
We then group those digit to form the equivalent number i.e. 110001110 10, dropping
the 0 of the most significant bit i.e. left hand end.
Assignment: Convert the following octal numbers to binary
1)
647
2)
834
3)
5867
HEXADECIMAL NUMBER SYSTEM:
The hexadecimal number system (Hex) uses base 16. The main purpose of this number
system is to represent the very large numbers of memory locations utilized by
microcontrollers and microprocessors. The Instructions that make up the assembly
language use to program microcontrollers and microprocessors also make use of the
hexadecimal number system. The hexadecimal number system use the number 0 to 9,
along with the letters A, B, C, D, E, and F, to make up the 16 symbols. The relationship
between hexadecimal, decimal and binary is shown in the figure below.
An important point to recognize about the hexadecimal number system is that 4 binary
digits represent a single hexadecimal digit. This is significant because a 16-bit binary
code can be represented with a 4 digit hexadecimal number.
The following table shows decimal number with hexadecimal and binary equivalents:-
Decimal
Hexadecimal
Binary
0
0
0000
1
1
0001
2
2
0010
3
3
0011
4
4
0100
5
5
0101
6
6
0110
7
7
0111
8
8
1000
9
9
1001
10
A
1010
11
B
1011
12
C
1100
13
D
1101
14
E
1110
15
F
1111
Digit Position - Each digit position in the hexadecimal number system represents a
power of 16, so that a number such as 183 (hexadecimal) is
162
161
16°
256
16
1
BINARY-TO-HEXADECIMAL CONVERSION
-
To convert binary
I 10 110 1100 11110 to hexadecimal, follows the following steps.
1.
Divide the binary number into groups of four bits starting from the least
significant LSD (right most digit) digit end. The above number becomes
110
2.
Next convert each group to the equivalent decimal number. The number in the
above example then becomes:
110
6
3.
1101 1001 1110
1101 1001 1110
13
9
14
The next step is to convert any number greater than 9 resulting from the above
step to the equivalent hexadecimal symbol. This becomes 6D 9E16
HEXADECIMAL-TO-BINARY CONVERSION - The reverse of the above rules
apply
when converting hexadecimal numbers to binary. E.g. to convert the number D39A
hexadecimal to binary we first convert the hexadecimal symbols to decimal so that the
number becomes:
D
3
9
A
13
3
9
10
We then write the binary equivalents for each of these numbers:
13
3
9
A
1101 0011 1001 1010
Finally, the numbers are grouped together to form the binary result, i.e.
1101001110011010.
SOLID STATE CIRCUIT TECHNOLOGY
These days, as is well known the idea is to make electrical components and their
associates circuits and controls as small and light as possible. This is called
miniaturization and of course the smaller and lighter you can make anything
(without compromising the necessary strength and integrity of components) that
goes into a flying machine the more space and/or weight is available for use as
payload. Along with this most desirable (for aircraft) trend miniaturization has also
brought with more efficiency and reliability and the concept of modular
construction, where almost any fault can be identified as occurring in a module and
that module can then be replaced. The faulty module is then returned to the
manufacturers for repair or trashing.
The advent of the printed circuit, the transistor and other semiconductor devices
and finally the solid state circuit has made this all possible. This revolution has also
meant a different and sometimes more complicated fields of learning has evolved,
some of which aircrew are required to be acquainted with in addition to the older
traditional DC/AC theory
A solid state circuit is a collection of components (such as diodes, transistors,
rectifiers, capacitors and resistors) which would normally be connected together as
separate discrete objects but instead are all "embedded" in micro-size sections of
semiconductor material. Over the years manufacturers of these micro-chips
(called ``chips" or circuit" packs") have managed to pack (integrate) more and
more "components" and circuitry into less and less semiconductor material, so you
now have a range of microchips from those considered to be small - scale integration
through medium and large scale to the more recent very Large - scale Integration
chips (SSI, MSI, LSI and VLSI). Most manufactures use a process called
Photolithography to achieve extremely small circuit. Photolithography imprints a
circuit on a silicon wafer by focusing a pattern of light into a concentrated area.
This process is similar to the one done in a dark room to imprint a negative
image on photographic paper chemical solvent are then used to etch the circuit
design into the silicon. By adding other materials to specific area of the silicon wafer
`doping' of the circuit is accomplished. A second layer of silicon may be added and
another combination of circuits produced within the IC. the silicon wafers are then
cut to size and assembled into the IC package.
These chips are able to perform a large number of individually dedicated functions
and by connecting certain chips, with known functions, together it is possible to
arrive at a "unit" having the characteristics you might require for any particular
job. Connecting these chips together (with other circuit devices) involves more
circuitry.
Although what a chip contains may be said to work in a highly complicated and
sophisticated way (just as old style using separate components joined together with
ordinary wire) the use of these micro-chips simplifies things a lot. One doesn't need
to know anything about the circuits and components inside the chip, only about the
connectors on the outside of the chip and the functions and parameters that are
associated with each connector.
Nowadays the operation of most of the electrical units inside aircraft is achieved
with the use of solid state technology. In order to control the operations the
controlling equipment has to make certain "logical" decisions based on the demand
input of the pilot and the positions of switches, solenoids and relays. Decisions by
equipment also have to be made in the case of fault conditions which are fully or
partly intended to be automatically operated equipment.
The elements that are required to make logical decisions are called LOGIC GATES.
A decision making circuit may have one logic gate only but it is more usual to find
several
or many in aircraft electrical situations depending on the complexity involved.
Similarly for a micro-chip, there may be a few or very many logic gates inside the
chip.
We are by nature familiar with logic, because our minds continuously use logic for
decision making function. We can solve mathematical problems, make rational
decision based on facts received, and modify our decisions as a result of new
information, or on previously acquired knowledge stored in our memory.
If
emotional and intuitive aspects are ruled out entirely, our mind is an approximation
of what digital logic does-electronically. Therefore, Digital Logic is a logical
decision-making process performed by electronic circuits.
Logical decisions-making process is the theoretical basis of digital electronic devices
such as computers, electronic calculators and digital clocks.
Logic gates
Logic gates or gates, are fundamental function performed by computers and related
equipment. A single integrated circuit (IC) within a computer contains several gates
circuit. Each gate may have several inputs and must have only one output. There
are six commonly used logic gates. The AND, the OR, the Invert (NOT), the NOR,
the NAND and the exclusive OR (ExOR). The name of each gate represents the
function it performs.
Truth Tables: are systematic means of displaying binary data. Truth tables
illustrate the relationship between a logic gates inputs and output. This type of data
display can be used to describe the operation of a gate or an IC. For troubleshooting
purpose, the truth table data for specific IC is often reviewed in order to determine
the correct output signal for a given set of inputs. Each logic gate have a symbol of
a specific shape. The symbols are designated to `point' in a given direction that is
the inputs are always listed on the left of the symbol and the output on the right.
Generally in the computer world `0' represents one state and '1' represents the other
(because the micro-chip inside the computer works only binary numbers) and with
a crossover into the world we experience `0' is the same as false and '1' means true.
Other expressions that are used:
1 = on, true, high (H), closed, engaged, positive voltage, negative voltage
0 = off, false, low (L) open, disengaged, negative voltage, positive voltage
In the electrical sense a positive voltage can represent l/true or 0/false and so can a
negative voltage. If a positive voltage is used to represent l/true then this is known
as POSITIVE logic, if a negative voltage then NEGATIVE logic. Positive and
negative are used in a relative sense. i.e. +.5 to + 2.5 (approx.) volts is often used as
one state and +3.5 to +5 volts using +.5 to +2.5 volts represent 1/true is negative logic
because +2 is negative relative to +3.5or +5.
NOTE:
Because the Logic gate works with only two states it is referred to as having a
BINARY function.
Because both the two states act at each of two input connectors the number of
possible input combinations is four (for the six basic gates we are considering.)
The AND Gate
Using positive logic then a voltage of +3.5 to +5 must be applied to both inputs A&B to
obtain an output voltage of +3.5 to +5. 1 f either input A or B or both has +.5 to +2
volts applied then output C would be +.5 to 2 volts. Using negative logic then a voltage
of +.5 to +2must be applied to both inputs A& B to obtain an output voltage of +.5 to
+2. If either input A or B (or both) has +4 to +5 volts applied then output C would be
+4 to +5 volts.
Figure 3 shows the symbol normally used to indicate a two-input AND gate, and also
a simple series electrical circuit which operates in the AND sense
Two switches in series are used to turn on a light. If both switches (inputs) are on (1),
the light will turn on. If either switch is off (0) (or both), then the light will be off (0).
If an AND gate has more than two input terminals then ALL inputs must be at (1) to
obtained a (1) output, and if only one input is at (0) then the output will be `0.'
The OR Gate
In Figure 4 below are shown the symbol for a two-input OR gate, the truth table and
another simple circuit (parallel) demonstrating the OR gate functioning
The OR gate is used to represent a situation where an input of (1) on either of the two
inputs A or B will produce a (1) output. Where there are more than two input terminal
then provided one has a (1) input then the output will be (1). Only if all inputs are at
(0) will the output be (0). The OR circuit at (b) shows two switches coming off of the
Busbar in parallel, and then connected to earth via a light. Both switches open (0) then
no light, and only one switch need be closed (1) to activate the light, though both closed
would also turn the light on.
The Exclusive OR (ExOR) Gate.
The purist looking at the truth table of OR logic could argue that it is not 100% OR
because it includes an AND case (A at (1) AND B at (1) produce (1) output. This gives
rise to the exclusive (of AND) OR logic, usually abbreviated to ExOR. The ExOR gate
is designed to produce a (1) output whenever its input signals are dissimilar, and a (0)
output for similar input signals.
Figure 6 (a) is the symbol for
ExOR logic function. Here
switch I is ganged (connected)
to another switch so that when
switch 1 is open the other
switch is closed and vice versa.
Switch 2 is also ganged and
works in a similar fashion.
If you follow through the circuit
you should be able to justify
the truth table in (c)
This arrangement would not normally be used for this function as it is far more
convenient and cheaper to use two AND gate together with one OR gate wired
up in a special way. i.e. the AND gate & the OR gate can be considered as two
very basic gates, a number of which can be wired up in different ways together
with another very basic logic gate called a NOT gate to obtained other types of
logic gates.
The NOT (Invert) gate.
The NOT gate is used simply to reverse the state (0 or 1) of the input signal.
A NOT circuit might be comprised of a switch controlling a relay that is normally
closed (light on). The relay turns OFF or ON a light. See Fig.7 (b)
if the switch (SWI) is turned ON the relay attracts the armature and the light
goes off.
The NAND Gate
NAND is a contraction of NOT AND because the NOT logic is combined with the
AND logic to form the NAND Gate i.e. The NAND gate is an AND Gate with an
inverted output. The output of this gate will be (1) if any inputs is (0) and this, of
course, is the exact opposite of an AND Gate. The symbol is shown in fig. 8
opposite, and note that a small empty circle is used to denote the invert (NOT)
logic.
The NOR Gate
The NOR (contraction of NOT OR) gate is an OR gate with an inverted output.
This results in a gate where any input being (1) will create a (0) output. The NOR
symbol and truth table
Are shown in Fig9.
In both the NAND and the NOR diagrams there is a short line drawn over the
letter C marking the outputs from these gates. This is often done to emphasize
the fact that there is an inversion of the normal signal and is in addition to the
small empty circle at the right end of the symbol.
Logic Gates in Circuit Wiring Diagram.
The modern trend is to do away with the older conventional wiring diagram with
its very many wires, junctions and discrete symbols and instead present a system
wiring diagram like the one in Fig. 10. At first sight it would seem difficult to
read, but with practice the
advantage of this type of presentation become obvious. The diagram in Fig. 10
(relatively simple) is that of a tricycle undercarriage system.
The three micro-switches on the left are normally open in flight (U/C UP) but are
closed when the appropriate U/C leg is down locked. The throttle micro-switch is open
for most of the range of movement of the throttle but closes when near to the shutthrottle position. There are two AND gates plus a NOR gate operating with positive
logic. The top AND gates' output terminal is connected through a light to negative
earth. The output will be 0 (light OFF) if any of the U/C switches are at0 OFF.
The output of the NOR gate will be 0 if any of the U/C switches are 1 (ON). Only if
ALL the U/C switches are 0 (OFF) will NOR gate output be 1. So all (U/C switches
must be OFF (U/C UP) and the throttle closed or almost closed for the lower AND gate
to receive a I on BOTH its input terminals. the horn will not come on unless both input
terminals of its AND gate are at I to produce a 1 (ON) at its output terminal. The horn
warns the pilot in an event he forgets to bring down and/or lock his landing gear.
BOOLEAN ALGEBRA
Boolean algebra is a form of algebra designed to deal specifically with two-state
functions used for the design of logic circuits. It allows the objectives of a logic circuit
to be stated in symbolic terms, so that the statement can then be equated with an actual
circuit. Boolean algebra was originated by English man George Boole (1815 - 1864) for
calculating rather than reasoning out problems of logical truth.
Boolean algebra differs from conventional algebra in that conventional algebra deals
with quantitative relationships, whereas Boolean algebra deals with logical
relationships. In conventional algebra we use symbolic quantities, such as X, Y, A, and
B to represent numbers. In solving conventional algebraic problems we are interested
in knowing the size of A, whether X is larger or smaller than Y or other information
related to quantity. In Boolean algebra we are interested only in knowing one of two
possible states of a symbolic term. For example, when we are using Boolean algebra
for digital logic, we are interested whether an algebraic term is I or 0. When it is used
for philosophical logic, a statement is said to be either true or false, Thus, Boolean
algebra is a two-valued system, uniquely different from conventional algebra.
DEFINITIONS AND POSTULATES
l.
The elements considered in Boolean algebra are called variables and are
generally represented by letter symbols: A,B,C,X,Y,W. etc. as with the
conventional algebra.
2.
Any variable or group of variables can have only one of two values either 1 or
0. The interpretation of 1 may be that the variable is present, is true, is a positive
voltage, or is in any other specifically defined state. A 0 then means that the
variable is absent, not true (false), is a negative voltage, or is in a state opposite
to that given to logic 1
In conventional algebra a dependent variable (y) may be some fiinetion of an
independent variable (x), written y = f (x). Similarly, in Boolean algebra a
dependent binary variable (C) may be a function of one or several independent
binary variables (A,B, etc), each of which may have a value of 0 or l. This is
written c = f (A,B, etc).
3.
The two ways of combining variables are as follows:
(a)
+
means logical addition (sum), which is read OR.
(b)
means logical multiplication (product), which is read AND.
The above are called logical connectives: they should not be confused with the +
and. signs of conventional algebra.
4.
The equality symbol = means that the variables or groups of variables on either
side of symbol are the same - both are in the same logical state
5.
The symbol - means negatation or a NOT function. If it is used with a variable,
the variable assumes the opposite sate. For example, IF A=1, then A =-0
BOOLEAN STATEMENTS
There are three kinds of Boolean statements:
l.
The AND statement represent the combination of variables by logic
multiplication. For example, C = A.B is an AND statement represented by the
AND gate symbol. The AND statement is read C equals A and B.
2.
The OR statement represents the combination of variables by logic additions for
example, C = A + B is an OR statement represented by the OR gate symbol. The
OR statement is read c equals A or B.
3.
A special form of the OR statement is the exclusive OR, which is read either A
or B, but not both. The exclusive OR statement is written C = A B and
represented by the excusive OR gate
Mixed Statements
A mixed statement consist of both AND statements and OR statements. It can be a sum
of products statement, or a product-of sums statement.
LAWS OF COMBINATION
The following basic laws are the basis of all simplification of Boolean equation.
1.
Law of Commutation or Commutative Law
The law of commutation states that it is immaterial which of several quantities
is taken first during addition and multiplication. For two logical variables, this
can be expressed by:
A + B = B + A------ (1)
A.B
2.
= B.A - -------- (2)
Law of Association or Associative Law
The law of association states that it is immaterial how the various
Terms of a logical expression are grouped together, or associated. For example:
A. (B. C)
_
(A.B).C
= A.B.C------------ (3)
A + (B+C)
_
(A+B) +C
= A+B+C----------- (4)
3.
Law of Distribution or Distributive Law states that a multiplier of a sum
operates on each term of the sum. For example, A. (B+C)=AB + A.C - ----(5)
conversely, the rule implies that common multipliers may be factored out of
sums, as in conventional algebra. For example, E = AC + AD + BC -+- BD =
A(C+D) + B(C + D) and factorizing again the common logical sum (CD), there
results E = (A + B). (C + D)
4.
Law of Tautology or Logical Relations.
Logical sum (OR)
A
B
A+B
0
0
0
0
1
1
1
0
1
1
1
1
We can establish the following relations by means of the above truth table.
i.
A + 0 = A. Using the truth table, if A = 0 then 0 +0 = 0
(first row), and if A = 1, then 1 + 0 = 1 (thirds row): hence the statement is
proved, similarly, we can show.
ii
A+1=1
iii
A+A=A
iv.
A+A= 1
Logical Product (AND)
A
B
A.B
0
0
0
0
1
0
1
0
0
1
1
1
The above truth table can also be used to establish the following relations.
v
A. 0 = 0
vi
A.1 = A
vii
A._A=A
viii
A.A=0
5.
Law of Double Complementation –
DEMORGAN'S THEOREM
Demorgan's Theorem - States that if the inversion bar is broken between two variables,
the symbol connecting the variables must be changed. The reverse is also true: if the
inversion is joined between two variables, the symbol connecting the variables must be
changed.
_
A+
C,
=
A + B . C~ (Using law of Double complementation)